Miss FOTA 32: Twice as Nice?

They all laughed when the province of Singletania announced that it was going to send a single-elimination candidate to the Miss FOTA 32 Pageant. How could a design with only one bracket be the fairest?

But they gasped when they got a look at Miss SE55779! They’d been expecting a frumpy design with fairness (C) near 100. But she piled lots of extra games into her svelte five rounds. Still, surely one could do better with a double-elimination tourney.

Better? Well, maybe, but there’s a price to pay.

We’ll start, as we did when we wanted a single-elimination candidate, with the simplest model. We know from previous research that the best 32DE is one that uses the late, or DE shift. Here’s the lower bracket: 32lowerde

This base bracket does a good deal better than the base SE bracket did, with f(C) at 78.93 rather than 97.24. But it also took almost twice as much time: 193 minutes to 99 minutes.

For the base SE bracket, we could make all rounds best of five, and still have a little time to spare. Here, it looks like we’d be pushing things making them all best of three. That bracket comes in at f(C) = 49.59 – a bit better than the SE bracket’s 50.82. But the best-of-three DE bracket also just flunks the time test, coming in at 503.9 minutes. So it’s still not clear that any DE is going to be better because we’re going to have to pare it back somewhere.

Before going there, let’s see if we might want to pare back a bit further to make room for a recharge round. At high-luck levels, we’ve seen recharges be modestly beneficial in a 16 bracket (in a very old post using a fairness (C) measure that’s been superseded), and pretty much a wash in a 64 bracket. We might fairly expect a small fairness (C) bonus with a 32. But adding a recharge to the best-of-three bracket not only exacerbates the time problem, running 533.8 minutes, but it does rather worse, with f(C) = 51.3. So there’s no reason to go that road.

It seems we’re going to have to make at least one round of the tourney a single game rather than best of three. With a double elimination base, there are more rounds to choose from – rounds A through M rather than A through E in the 32SE. But, diving into the flow simulation, it becomes apparent that not all of the rounds are good candidates for trimming. Rounds B and F run together, as do rounds C and G. To save much time, we’d have to cut both B and F (or both C and G) to get any meaningful time savings.

Assuming, then, that we want to cut as far as possible from the finals, we have two sensible choices. Cut round A, affecting everyone, or round H, affecting only four 2-1 teams in the lower bracket, but hitting teams that are nearer the prize money. Trimming round A saves more time than trimming round H, with times of 455 minutes and 475 minutes, respectively. But it also means a bigger hit to f(C): 52.44 to 50.24. Since we don’t need much time, we opt for trimming the H round.

But looking at the average wait times suggests that there’s a round where we can add games without slowing the tourney as a whole. The team that loses E1 waits, on average, waits a little over an hour for its next match in round K. So, making round E best of five costs almost nothing in terms of total time.

Bingo. The bracket that’s best of three everywhere except for single games in round H and best of five in round E comes in at f(C) = 49.2 – we gained more from E than we lost from H. And there’s still some time left, with T = 479.7.

Looking carefully at the wait times, it looks like we might just be able to squeeze in another best of five, this time in round C. It works, but only barely. f(C) = 47.77, and T = 493.9.

That’s our candidate. We’ll name her Miss 32DEhCD, mentioning only the altered rounds so that her name isn’t too long. This is, to be sure, a format that would raise eyebrows in any real tournament. It would be hard to explain why rounds C and E were best of five while round H was a single game.

In Slow, Shifty Brackets, we saw that adding games to the upper bracket can consume all of the time savings from a bracket shift, and then some. We get away with it here only because we’re exceptionally judicious in choosing the one round we’ll trip and the two rounds we’ll augment.

Miss DE32hCD beats Miss SE55779 by a solid three points of fairness (C), but I’m not sure she’s really the winner. SE55779 is a format that I’d be happy to run in a real tourney, but I wouldn’t dare try to run a DE32hCD. And no format that I might be willing to run does better than SE55779.

Before making putting the pageant judges to the choice, we’ll see if there’s anything else they might want to consider. Is it possible to get some value from a third bracket?

Good question. I think it would be like omitting the recharge round in reverse, denying a chance to the lower bracket winner rather than the upper bracket winner. At luck = zero it would make no difference, but I think it would hurt fairness (C) increasingly as the luck factor increases. I’ll run a simulation or two when I get the leisure to see if I’m guessing right, but don’t hold your breath–I’m pretty backed up just now.